A002582 -id:A002582 - cp's OEIS frontend

A033312 a(n) = n! - 1.

0, 0, 1, 5, 23, 119, 719, 5039, 40319, 362879, 3628799, 39916799, 479001599, 6227020799, 87178291199, 1307674367999, 20922789887999, 355687428095999, 6402373705727999, 121645100408831999, 2432902008176639999, 51090942171709439999, 1124000727777607679999

Offset: 0

N. J. A. Sloane. This sequence appeared in the 1973 "Handbook", but was then dropped from the database. Resubmitted by Eric W. Weisstein. Entry revised by N. J. A. Sloane, Jun 12 2012

a(n) gives the index number in any table of permutations of the entry in which the last n + 1 items are reversed. - Eugene McDonnell (eemcd(AT)mac.com), Dec 03 2004
a(n), n >= 1, has the factorial representation [n - 1, n - 2, ..., 1, 0]. The (unique) factorial representation of a number m from {0, 1, ... n! - 1} is m = sum(m_j(n)*j!, j = 0 .. n - 1) with m_j(n) from {0, 1, .., j}, n>=1. This is encoded as [m_{n-1},m_{n-2},...,m+1,m_0] with m_0=0. This can be interpreted as (D. N.) Lehmer code for the lexicographic rank of permutations of the symmetric group S_n (see the W. Lang link under A136663). The Lehmer code [n - 1, n - 2, ..., 1, 0] stands for the permutation [n, n - 1, ..., 1] (the last in lexicographic order). - Wolfdieter Lang, May 21 2008
For n >= 3: a(n) = numbers m for which there is one iteration {floor (r / k)} for k = n, n - 1, n - 2, ... 2 with property r mod k = k - 1 starting at r = m. For n = 5: a(5) = 119; floor (119 / 5) = 23, 119 mod 5 = 4; floor (23 / 4) = 5, 23 mod 4 = 3; floor (5 / 3) = 1, 5 mod 3 = 2; floor (1 / 2) = 0; 1 mod 2 = 1. - Jaroslav Krizek, Jan 23 2010
For n = 4, define the sum of all possible products of 1, 2, 3, 4 to be 1 + 2 + 3 + 4 add 1*2 + 1*3 + 1*4 add 2*3 + 2*4 + 3*4 add 1*2*3 + 1*2*4 + 1*3*4 + 2*3*4 add 1*2*3*4. The sum of this is 119 = (4 + 1)! - 1. For n = 5 I get the sum 719 = (5 + 1)! - 1. The proof for the general case seems to follow by induction. - J. M. Bergot, Jan 10 2011

			G.f. = x^2 + 5*x^3 + 23*x^4 + 119*x^5 + 719*x^6 + 5039*x^7 + 40319*x^8 + ...

Arthur T. Benjamin and Jennifer J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, identity 181, p. 92.
Michael Doob, The Canadian Mathematical Olympiad & L'Olympiade Mathématique du Canada 1969-1993, Canadian Mathematical Society & Société Mathématique du Canada, Problem 6, 1969, p. 3, 1993.
Problem 598, J. Rec. Math., 11 (1978), 68-69.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Jonathan Beagley and Lara Pudwell, Colorful Tilings and Permutations, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.4.
The IMO Compendium, Problem 6, 1st Canadian Mathematical Olympiad 1969.
Stéphane Legendre and Philippe Paclet, On the Permutations Generated by Cyclic Shift , J. Int. Seq. 14 (2011) # 11.3.2.
Gerard P. Michon, Wilson's Theorem.
Hisanori Mishima, Factorizations of many number sequences.
Hisanori Mishima, Factorizations of many number sequences.
Michael Penn, Make it look like a simple calculus problem., YouTube video, 2021.
Andrew Walker, Factors of n! +- 1.
Eric Weisstein's World of Mathematics, Factorial.
Eric Weisstein's World of Mathematics, Permutation Pattern.
Index entries for sequences related to factorial base representation.
Index entries for sequences related to factorial numbers.
Index to sequences related to Olympiads.

Cf. A000142, A001563 (first differences), A002582, A002982, A038507 (factorizations), A054415, A056110, A331373.

Row sums of A008291.

[Factorial(n)-1: n in [0..25]]; // Vincenzo Librandi, Jul 20 2011

FoldList[#1*#2 + #2 - 1 &, 0, Range[19]] (* Robert G. Wilson v, Jul 07 2012 *)
Range[0, 19]! - 1 (* Alonso del Arte, Jan 24 2013 *)

A033312(n):= n!-1$
makelist(A033312(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

a(n)=n!-1 \\ Charles R Greathouse IV, Jul 19 2011

a(n) = Sum_{k = 1 .. n} (k-1)*(k-1)!.
a(n) = a(n - 1)*(n - 1) + a(n - 1) + n - 1, a(0) = 0. - Reinhard Zumkeller, Feb 03 2003
a(0) = a(1) = 0, a(n) = a(n - 1) * n + (n - 1) for n >= 2. - Jaroslav Krizek, Jan 23 2010
E.g.f.: 1/(1 - x) - exp(x). - Sergei N. Gladkovskii, Jun 29 2012
0 = 1 + a(n)*(+a(n+1) - a(n+2)) + a(n+1)*(+3 + a(n+1)) + a(n+2)*(-1) for n>=0. - Michael Somos, Feb 24 2017
Sum_{n>=2} 1/a(n) = A331373. - Amiram Eldar, Nov 11 2020

A002583 Largest prime factor of n! + 1.

Original entry on oeis.org

2, 2, 3, 7, 5, 11, 103, 71, 661, 269, 329891, 39916801, 2834329, 75024347, 3790360487, 46271341, 1059511, 1000357, 123610951, 1713311273363831, 117876683047, 2703875815783, 93799610095769647, 148139754736864591, 765041185860961084291, 38681321803817920159601

Offset: 0

N. J. A. Sloane

Theorem: For any N, there is a prime > N. Proof: Consider any prime factor of N!+1.
Cf. Wilson's theorem (1770): p | (p-1)! + 1 iff p is a prime.
If n is in A002981, then a(n) = n!+1. - Chai Wah Wu, Jul 15 2019

			(0!+1)=[2], (1!+1)=[2], (2!+1)=[3], (3!+1)=[7], (4!+1)=25=5*[5], (5!+1)=121=11*[11], (6!+1)=721=7*[103], (7!+1)=5041=71*[71], etc. - Mitch Cervinka (puritan(AT)toast.net), May 11 2009

M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Georg Fischer, Table of n, a(n) for n = 0..139 (first 101 terms originally derived from Hisanori Mishima's data by T. D. Noe)
A. Borning, Some results for k!+-1 and 2.3.5...p+-1, Math. Comp., 26 (1972), 567-570.
P. Erdős and C. L. Stewart, On the greatest and least prime factors of n! + 1, J. London Math. Soc. (2) 13:3 (1976), pp. 513-519.
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy]
Li Lai, On the largest prime divisor of n! + 1, arXiv:2103.14894 [math.NT], 2021.
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
H. P. Robinson and N. J. A. Sloane, Correspondence, 1971-1972
Blake C. Stacey, Equiangular Lines, Ch. 1, A First Course in the Sporadic SICs, SpringerBriefs in Math. Phys. (2021) Vol. 41, see page 5.
R. G. Wilson v, Explicit factorizations

Cf. A002582, A002981, A038507, A051301, A056111, A096225.

[Maximum(PrimeDivisors(Factorial(n)+1)): n in [0..30]]; // Vincenzo Librandi, Feb 14 2020

PrimeFactors[n_]:=Flatten[Table[ #[[1]],{1}]&/@FactorInteger[n]]; Table[PrimeFactors[n!+1][[ -1]],{n,0,35}] ..and/or.. Table[FactorInteger[n!+1,FactorComplete->True][[ -1,1]],{n,0,35}] (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)
FactorInteger[#][[-1,1]]&/@(Range[0,30]!+1) (* Harvey P. Dale, Sep 04 2017 *)

a(n)=my(f=factor(n!+1)[,1]);f[#f] \\ Charles R Greathouse IV, Dec 05 2012

Erdős & Stewart show that a(n) > n + (1-o(1))log n/log log n and lim sup a(n)/n > 2. - Charles R Greathouse IV, Dec 05 2012

Lai proves that lim sup a(n)/n > 7.238. - Charles R Greathouse IV, Jun 22 2021

More terms from Robert G. Wilson v, Aug 01 2000

Corrected by Jud McCranie, Jan 03 2001

A051301 Smallest prime factor of n!+1.

Original entry on oeis.org

2, 2, 3, 7, 5, 11, 7, 71, 61, 19, 11, 39916801, 13, 83, 23, 59, 17, 661, 19, 71, 20639383, 43, 23, 47, 811, 401, 1697, 10888869450418352160768000001, 29, 14557, 31, 257, 2281, 67, 67411, 137, 37, 13763753091226345046315979581580902400000001

Offset: 0

Labos Elemer

nonn

Theorem: For any N, there is a prime > N. Proof: Consider any prime factor of N! + 1.
Cf. Wilson's Theorem (1770): p | (p-1)! + 1 if and only if p is a prime.
If n is in A002981, then a(n) = n!+1. - Chai Wah Wu, Jul 15 2019

			a(3) = 7 because 3! + 1 = 7.
a(4) = 5 because 4! + 1 = 25 = 5^2. (5! + 1 is also the square of a prime).
a(6) = 7 because 6! + 1 = 721 = 7 * 103.

Albert H. Beiler, "Recreations in The Theory of Numbers, The Queen of Mathematics Entertains," Dover Publ. NY, 1966, Page 49.
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors).

Chai Wah Wu, Table of n, a(n) for n = 0..138 n = 0..100 derived from Hisanori Mishima's data by T. D. Noe.
A. Borning, Some results for k!+-1 and 2.3.5...p+-1, Math. Comp., 26 (1972), 567-570.
P. Erdős and C. L. Stewart, On the greatest and least prime factors of n! + 1, J. London Math. Soc. (2) 13:3 (1976), pp. 513-519.
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy]
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
R. G. Wilson v, Explicit factorizations

Cf. A002583, A002981, A038507, A096225.

with(numtheory): A051301 := n -> sort(convert(divisors(n!+1),list))[2]; # Corrected by Peter Luschny, Jul 17 2009

Do[ Print[ FactorInteger[ n! + 1, FactorComplete -> True ] [ [ 1, 1 ] ] ], {n, 0, 38} ]
FactorInteger[#][[1,1]]&/@(Range[0,40]!+1) (* Harvey P. Dale, Oct 16 2021 *)

a(n)=factor(n!+1)[1,1] \\ Charles R Greathouse IV, Dec 05 2012

Erdős & Stewart show that a(n) > n + (l-o(l))log n/log log n except when n + 1 is prime, and that a(n) > n + e(n)sqrt(n) for almost all n where e(n) is any function with lim e(n) = 0. - Charles R Greathouse IV, Dec 05 2012

By Wilson's theorem, a(n) >= n + 1 with equality if and only if n + 1 is prime. - Chai Wah Wu, Jul 14 2019

A002585 Largest prime factor of 1 + (product of first n primes).

Original entry on oeis.org

3, 7, 31, 211, 2311, 509, 277, 27953, 703763, 34231, 200560490131, 676421, 11072701, 78339888213593, 13808181181, 18564761860301, 19026377261, 525956867082542470777, 143581524529603, 2892214489673, 16156160491570418147806951, 96888414202798247, 1004988035964897329167431269

Offset: 1

N. J. A. Sloane

Based on Euclid's proof that there are infinitely many primes.

The products of the first primes are called primorial numbers. - Franklin T. Adams-Watters, Jun 12 2014

M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors).
M. Kraitchik, Introduction à la Théorie des Nombres. Gauthier-Villars, Paris, 1952, p. 2.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Table of n, a(n) for n = 1..98
A. Borning, Some results for k!+-1 and 2.3.5...p+-1, Math. Comp., 26 (1972), 567-570.
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy]
S. Kravitz and D. E. Penney, An extension of Trigg's table, Math. Mag., 48 (1975), 92-96.
S. Kravitz and D. E. Penney, An extension of Trigg's table, Math. Mag., 48 (1975), 92-96. [Annotated scanned copy; also letter from N. J. A. Sloane to John Selfridge]
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018. - From _N. J. A. Sloane_, Jun 13 2012
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
John Selfridge, Marvin Wunderlich, Robert Morris, N. J. A. Sloane, Correspondence, 1975
Eric Weisstein's World of Mathematics, Euclid Number.
R. G. Wilson v, Explicit factorizations

Cf. A002110, A002584, A006530, A006862, A051342.

FactorInteger[#][[-1,1]]&/@Rest[FoldList[Times,1,Prime[Range[30]]]+1] (* Harvey P. Dale, Apr 10 2012 *)

a(n)=my(f=factor(prod(i=1,n,prime(i))+1)[,1]); f[#f] \\ Charles R Greathouse IV, Feb 07 2017

a(n) = A006530(A006862(n)). - Amiram Eldar, Feb 13 2020

More terms from Labos Elemer, May 02 2000
More terms from Robert G. Wilson v, Mar 24 2001
Terms up to a(81) in b-file added by Sean A. Irvine, Apr 19 2014
Terms a(82)-a(87) in b-file added by Amiram Eldar, Feb 13 2020
Terms a(88)-a(98) in b-file added by Max Alekseyev, Aug 26 2021

A051342 Smallest prime factor of 1 + (product of first n primes).

Original entry on oeis.org

3, 7, 31, 211, 2311, 59, 19, 347, 317, 331, 200560490131, 181, 61, 167, 953, 73, 277, 223, 54730729297, 1063, 2521, 22093, 265739, 131, 2336993, 960703, 2297, 149, 334507, 5122427, 1543, 1951, 881, 678279959005528882498681487, 87549524399, 23269086799180847

Offset: 1

Labos Elemer

nonn

Based on Euclid's proof that there are infinitely many primes.

Amiram Eldar, Table of n, a(n) for n = 1..87
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy]
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
R. G. Wilson v, Explicit factorizations

Cf. A014545, A002585.

a:= proc(n)
local N, F, i;
  N:= 1 + mul(ithprime(i),i=1..n);
  F:= select(type,map(t->t[1],ifactors(N,easy)[2]),integer);
if nops(F) >= 1 then return min(F) fi;
  min(map(t->t[1],ifactors(N)[2]))
end proc:
seq(a(n),n=1..40); # Robert Israel, Oct 19 2014

Map[FactorInteger,
   Table[Product[Prime[n], {n, 1, m}] + 1, {m, 1, 36}]][[All,
1]][[All, 1]] (* Geoffrey Critzer, Oct 19 2014 *)
FactorInteger[#][[1,1]]&/@(FoldList[Times,Prime[Range[40]]]+1) (* Harvey P. Dale, Oct 08 2021 *)

a(n) = factor(1 + prod(i=1, n, prime(i)))[1, 1]; \\ Michel Marcus, Dec 10 2013

a(n) = A020639(1+A002110(n)).

One more term from Michel Marcus, Dec 10 2013

A104359 Greatest prime factor of A104357(n) = A104350(n) - 1.

Original entry on oeis.org

1, 5, 11, 59, 179, 1259, 229, 7559, 37799, 415799, 17569, 71437, 18979, 62597, 1135133999, 1646947, 445771, 277021, 5499724229999, 2217247573, 721381, 46313123, 29220034833989999, 16347569521, 5464930609, 4939567, 319699160368361, 2605998587146349, 178974179, 15701603, 116318025830291273, 126202964557

Offset: 2

Reinhard Zumkeller, Mar 06 2005

nonn

Tyler Busby, Table of n, a(n) for n = 2..159 (terms 2..74 from Amiram Eldar, terms 75..145 from Max Alekseyev)
Reinhard Zumkeller, Products of largest prime factors of numbers <= n

Cf. A006530, A002582, A002584, A104350, A104357, A104358, A104360, A104361, A104362, A104363, A104364, A104367.

a[n_] := FactorInteger[-1 + Product[FactorInteger[k][[-1, 1]], {k, 1, n}]][[-1, 1]]; Array[a, 50, 2] (* Amiram Eldar, Feb 12 2020 *)

gpf(n) = if (n==1, 1, vecmax(factor(n)[,1])); \\ A006530
a(n) = gpf(prod(i=2, n, gpf(i))-1); \\ Michel Marcus, Feb 21 2023

a(n) = A006530(A104357(n)).

A002584 Largest prime factor of product of first n primes - 1, or 1 if no such prime exists.

Original entry on oeis.org

1, 5, 29, 19, 2309, 30029, 8369, 929, 46027, 81894851, 876817, 38669, 304250263527209, 92608862041, 59799107, 1143707681, 69664915493, 1146665184811, 17975352936245519, 2140320249725509

Offset: 1

N. J. A. Sloane

The products of the first primes are called primorial numbers. - Franklin T. Adams-Watters, Jun 12 2014

M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors).
M. Kraitchik, Introduction à la Théorie des Nombres. Gauthier-Villars, Paris, 1952, p. 2.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Sean A. Irvine and Amiram Eldar, Table of n, a(n) for n = 1..99 (terms 1..91 from Sean A. Irvine)
A. Borning, Some results for k!+-1 and 2.3.5...p+-1, Math. Comp., 26 (1972), 567-570.
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy]
S. Kravitz and D. E. Penney, An extension of Trigg's table, Math. Mag., 48 (1975), 92-96.
S. Kravitz and D. E. Penney, An extension of Trigg's table, Math. Mag., 48 (1975), 92-96. [Annotated scanned copy; also letter from N. J. A. Sloane to John Selfridge]
Hisanori Mishima, Factorizations of many number sequences
John Selfridge, Marvin Wunderlich, Robert Morris, N. J. A. Sloane, Correspondence, 1975
R. G. Wilson v, Explicit factorizations.

Cf. A002110, A002585, A006530, A057588.

Prepend[Table[ Max[Transpose[FactorInteger[(Times @@ Prime[Range[i]]) - 1]][[1]]], {i, 2, 20}], 1]
FactorInteger[#][[-1,1]]&/@Rest[FoldList[Times,1,Prime[Range[20]]]-1] (* Harvey P. Dale, Feb 27 2013 *)

a(n)=if(n>1, my(f=factor(prod(i=1,f,prime(i)))[,1]); f[#f], 1) \\ Charles R Greathouse IV, Feb 08 2017

a(n) = A006530(A057588(n)). - Amiram Eldar, Feb 13 2020

More terms from J. L. Selfridge

Further terms from Labos Elemer, Oct 25 2000

A064145 a(n) = tau(n!-1) or number of divisors of n!-1.

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 4, 6, 4, 16, 2, 4, 2, 24, 4, 8, 8, 8, 4, 16, 8, 4, 4, 8, 4, 4, 16, 32, 2, 8, 2, 2, 4, 8, 4, 32, 2, 16, 4, 16, 16, 128, 16, 32, 32, 4, 16, 8, 4, 32, 32, 16, 64, 64, 32, 64, 32, 4, 8, 16, 16, 32, 16, 64, 16, 128, 4, 64, 32, 32, 8, 16, 32, 128, 8

Offset: 2

Vladeta Jovovic, Sep 11 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 2..135
Hisanori Mishima, Factorizations of many number sequences

Cf. A000005, A002582, A027423, A002981, A002982, A064144.

Do[ Print[ DivisorSigma[0, n! - 1]], {n, 2, 40} ]
DivisorSigma[0,Range[2,80]!-1] (* Harvey P. Dale, Aug 17 2024 *)

{ f=1; for (n=2, 100, f*=n; if (n>1, a=numdiv(f - 1), a=0); write("b064145.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 09 2009

More terms from Robert G. Wilson v, Oct 04 2001
a(51)-a(76) from Harry J. Smith, Sep 09 2009
Ambiguous term a(1) removed by Max Alekseyev, May 06 2022

A057713 Smallest prime divisor of Kummer numbers ( = primorials - 1), or 1 if no such prime exists.

Original entry on oeis.org

1, 5, 29, 11, 2309, 30029, 61, 53, 37, 79, 228737, 229, 304250263527209, 141269, 191, 87337, 27600124633, 1193, 163, 260681003321, 313, 163, 139, 23768741896345550770650537601358309, 66683, 2990092035859, 15649, 17515703, 719, 295201, 15098753, 10172884549, 20962699238647, 4871, 673, 311, 1409, 1291, 331, 1450184819, 23497, 711427, 521, 673, 519577, 1372062943, 56543, 811, 182309, 53077, 641, 349, 389

Offset: 1

Labos Elemer, Oct 25 2000

nonn

			6th term in the sequence corresponds to 7th primorial = 510510 and 510509 = 61 * 8369, so a(7) = 61.

Amiram Eldar, Table of n, a(n) for n = 1..99
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy]
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018. - _N. J. A. Sloane_, Jun 13 2012
Hisanori Mishima, Factorizations of many number sequences
R. G. Wilson v, Explicit factorizations

Cf. A002110, A002584, A002585, A006862, A020639, A057588.

Map[If[PrimeQ@ #, #, FactorInteger[#][[1, 1]]] &, FoldList[#1 #2 &, Prime@ Range@ 36] - 1] (* Michael De Vlieger, Feb 18 2017 *)

a(n) = A020639(A057588(n)). - Amiram Eldar, Feb 13 2020

More terms from Klaus Brockhaus, Larry Reeves (larryr(AT)acm.org) and Robert G. Wilson v, Apr 02 2001

A054415 Smallest prime factor of n!-1 (for n>2), a(2)=1.

Original entry on oeis.org

1, 5, 23, 7, 719, 5039, 23, 11, 29, 13, 479001599, 1733, 87178291199, 17, 3041, 19, 59, 653, 124769, 23, 109, 51871, 625793187653, 149, 20431, 29, 239, 31, 265252859812191058636308479999999, 787, 263130836933693530167218012159999999, 8683317618811886495518194401279999999

Offset: 2

Henry Bottomley, May 10 2000

nonn

The initial term a(2)=1 is not a prime, but it does not affect search results and may prevent submission of duplicates. - M. F. Hasler, Oct 31 2012

			a(3)=5 because 3!-1=5 which is prime; a(5)=7 because 5!-1=119=7*17 and 7<17

Chai Wah Wu, Table of n, a(n) for n = 2..153 (n = 2..135 from Amiram Eldar)
P. Erdős and C. L. Stewart, On the greatest and least prime factors of n! + 1, J. London Math. Soc. (2) 13:3 (1976), pp. 513-519.
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy]
Hisanori Mishima, Factorizations of many number sequences: n! - 1 (n = 1 to 100); Primorials - 1.
R. G. Wilson v, Explicit factorizations

Cf. A002582, A033312, A051301.

Do[ Print[ FactorInteger[ n! - 1, FactorComplete -> True][ [1, 1] ] ], {n, 3, 32} ]

A054415(n)=if(n>2,factor(n!-1)[1,1],1)  \\ M. F. Hasler, Oct 31 2012

Erdős & Stewart show that a(n) > n + (l-o(l))log n/log log n except when n+1 is prime, and that a(n) > n + e(n)sqrt(n) for almost all n where e(n) is any function with lim e(n) = 0. - Charles R Greathouse IV, Dec 05 2012

More terms from Robert G. Wilson v, Aug 01 2000

More terms from Amiram Eldar, Oct 07 2019

cp's OEIS Frontend

A033312 a(n) = n! - 1.

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A002583 Largest prime factor of n! + 1.

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A051301 Smallest prime factor of n!+1.

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A002585 Largest prime factor of 1 + (product of first n primes).

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A051342 Smallest prime factor of 1 + (product of first n primes).

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A104359 Greatest prime factor of A104357(n) = A104350(n) - 1.

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